Recently, substantial attention has been paid to improve the spectral efficiency of communication setups using different spectrum sharing techniques. This paper aims to study the capacity of Rayleigh-fading spectrum sharing channels in the case where there is correlation between the fading channels. Assuming perfect channel state information, the channel capacity is obtained under both adaptive and nonadaptive unlicensed user power allocation capabilities and the licensed user outage probability and received interference power constraints. Theoretical and simulation results show that (1) there is considerable potential for data transmission of unlicensed users with limited degradation of the licensed users' data transmission efficiency, (2) while the effect of fading channels dependencies is ignorable at low correlation values, the channel capacity is significantly affected as the channels correlation increases, and (3) depending on the fading distributions, the unlicensed user channel capacity is not essentially a decreasing function of the licensed user transmission power in the licensed user outage-limited scenario.